Yes, the Bill of Rights--and the entire Constitution--exists to "hamstring the government." But not completely--which is the issue here. The only way to "hamstring the government" completely would be to have no "government" at all (where 'government' is defined as a monopoly provider of justice and security.)
Of course, as a Voluntaryist, I strongly advocate for a strictly voluntary society with no large- and/or wide-scale monopolies at all--especially not a monopoly in the provision of justice and security. That said, Constitutionally there's a fundamental difference between a government official making his case for or against the actions, statements or decisions of private persons or of other entities versus making implied threats to punish those who fail to act as desired, or who just disagree. The Federal trial court in the case at hand (which the courts often refer to as the "trier of facts,") and also the Federal appellate court, both found that the government had crossed that line.
As Justice Alito noted, the Supreme Court is not supposed to override the findings of fact of the courts whose decisions are being appealed to SCOTUS. The evidence presented to the "trier of fact" very convincingly showed that the statements and actions of the government officials involved to the social media companies (and others) were of the form "Nice samizdat publishing platform ya' got there. Be a shame if something happened to it..."
If an Emperor, King or mob boss "requested" that you do—or not do—something, would you react to that request as though there might be an implied "or else" to it?
Constitutionally, what matters here is the 'State Actor Doctrine,' and also the legal precedents that hold that private entities who act with legal immunity (granted by the state) are state actors when they perform actions for which the state's laws give them immunity.
Key fact: Section 230 of the Communications Decency Act immunizes social media companies for any censorship actions they take for the purpose of censoring "offensive material." That means that social media companies, when they censor based on "community standards," are legally "state actors" according to current binding SCOTUS precedents. And the First Amendment absolutely applies to state actors.
See "Skinner v. Railway Labor Executives’ Assoc." for details.
Note: 'Samizdat' is a Russian term coined back when the Soviet Communists were censoring speech. Literally, it means "self publishing." It referred to the underground publishing of news and opinion in the Soviet Union that the authorities were not successful in completely shutting down...although they tried.
